The Fourier transform of a complex Gaussian pulse is derived in
§D.8 of Appendix D:

(11.27)

This result is valid when
is complex.
Writing
in terms of real variables
and
as

(11.28)

we have

(11.29)

That is, for
complex,
is a chirplet (Gaussian-windowed
chirp). We see that the chirp oscillation frequency is zero at time
. Therefore, for signal modeling applications, we typically add
in an arbitrary frequency offset at time 0, as described in the next
section.